A real function is called delta-convex if it can be expressed as a difference of two convex functions. A subset of a Euclidean space is called WDC if it is a sublevel set of a delta-convex function at a weakly regular value. The sets of positive reach, for instance, form a strict subclass of WDC sets. In the talk, the existence and properties of Federer's curvature measures, namely the validity of kinematic formulas, for the class of WDC sets will be discussed. The presented results are a joint work with Jan Rataj and Joseph Fu.

We will give some sample results from the new higher-dimensional theory of complex fractal dimensions developed jointly with Goran Radunovic and Darko Zubrinic in the forthcoming 530-page research monograph (joint with these same co-authors), "Fractal Zeta Functions and Fractal Drums: Higher Dimensional Theory of Complex Dimensions" (Springer, 2016). We will also explain its connections with the earlier one-dimensional theory of complex dimensions developed, in particular, in the research monograph (by M. L. Lapidus and M. van Frankenhuijsen) entitled "Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings" (Springer Research Monographs, Springer, New York, 2013; 2nd rev. and enl. edn. of the 2006 edn.).

In particular, to an arbitrary compact subset A of the N-dimensional Euclidean space (or, more generally, to any relative fractal drum), we will associate new distance and tube zeta functions, as well as discuss their basic properties, including their holomorphic and meromorphic extensions, and the nature and distribution of their poles (or 'complex dimensions'). We will also show that the abscissa of convergence of each of these fractal zeta functions coincides with the upper box (or Minkowski) dimension of the underlying compact set A, and that the associated residues are intimately related to the (possibly suitably averaged) Minkowski content of A. Example of classical fractals and their complex dimensions will be provided. Finally, if time permits, we will discuss and extend to any dimension the general definition of fractality proposed by the authors (and M-vF) in their earlier work, as the presence of nonreal complex dimensions. We will also provide examples of "hyperfractals", for which the 'critical line' {Re(s)=D}, where D is the Minkowski dimension, is not only a natural boundary for the associated fractal zeta functions, but also consist entirely of singularities of those zeta functions.

These results are used, in particular, to show the sharpness of an estimate obtained for the abscissa of meromorphic convergence of the spectral zeta functions of fractal drums.

Furthermore, we will also briefly discuss recent joint results in which we obtain general fractal tube formulas in this context (that is, for compact subsets of Euclidean space or for relative fractal drums), expressed in terms of the underlying complex dimensions.

We may close with a brief discussion of a few of the many open problems stated at the end of the aforementioned forthcoming book (and in the accompanying series of seven papers, joint with the same authors).

The design problem of finding one or more reflecting or refracting surfaces to redistribute the light emitted from a source into a prescribed illumination pattern can be stated as an inverse problem. For various design tasks, it has been shown that solutions can be constructed as envelopes of elementary building blocks such as paraboloids, ellipsoids or hyperboloids. Moreover, the methods employed in the analysis can often be motivated in analogy to methods from convex analysis and optimal transport theory. I will present an inverse problem arising in the design of a free-form reflector surface for a collimated source. Aside from a geometric proof of existence and uniqueness, I plan to outline an interesting variational approach. The latter is closely linked to optimal transport theory and its previous applications to lens and reflector design problems.

Variational problems involving the m dimensional area integrand in n dimensional Euclidean space lead naturally to various classes of m dimensional surfaces with singularities; all of which give rise to integral varifolds. The subsequent regularity theory is strongly tied to elliptic partial differential equations. Based on this connection, the talk will systematically explore a series of results - both positive and negative - investigating which of the familiar a priori estimates and regularity results of elliptic PDE have analogous formulations for integral varifolds. Some of these results are joint work with Sławomir Kolasiński.

3. DezemberMichael Hinz (University Bielefeld)"An introduction to the analysis on fractals and some recent results"

Abstract: The talk deals with analysis and stochastic processes on fully singular spaces (i.e. singular at every or almost every point). In the first part we give a brief introduction to the subject which was started in the late eighties and early nineties by Goldstein, Kusuoka, Barlow, Bass, Kigami and others and is now referred to as 'Analysis on fractals'. Markov processes and their energy functionals (Dirichlet forms) play a key role. In the second part we explain items of a related vector calculus and point out some recent results and applications.

Abstract: A covering graph of a finite graph whose covering transformation group is abelian is called crystal lattice. Around 2000, Kotani, Shirai and Sunada have studied long time behavior of symmetric random walks on crystal lattices. In this talk, we discuss about the non-symmetric random walks on crystal lattices. This talk is based on a joint work with Hiroshi Kawabi(Okayama) and Motoko Kotani(Tohoku).

Abstract: Weyl's law describes the asymptotic growth of the eigenvalues of the Laplacian on a closed manifold or a smooth bounded domain. We consider such laws for commutators $[P,f]$ and related Hankel operators for a pseudodifferential operator $P$ and a function $f$, in cases where the singularities of $f$ govern the growth of the eigenvalues. Their study is based on an interplay of techniques from spectral theory, harmonic analysis on $\mathbb{R}^n$ and the Heisenberg group, and noncommutative geometry.
In this talk we illustrate the rich spectral asymptotics of $[P,f]$, when $P$ is the Szegö projection on $S^1$, the truncation to positive Fourier modes. Higher-dimensional variants, and the general theory, lead to applications in complex analysis and noncommutative sub-Riemannian geometry. (joint work with Magnus Goffeng)

18. Juni (Two talks 16.00-18.00)Radoslaw Wojciechowski (City University of New York)The Feller Property for GraphsAbstract: The Feller property concerns the behavior at infinity of solutions of the heat equation. In this talk, we will follow recent work of Pigola and Setti who use a maximum principle approach to derive certain criteria for the Feller property on Riemannian manifolds. In particular, we will discuss this in the spherically symmetric case.Ivan Veselic (TU Chemnitz)Uncertainty and unique continuation principles for the observation of eigenfunctions

21. MaiSergei Matveev (Chelyabinsk State University)The Diamond Lemma and prime decompositionsWe describe a far generalization of the famous Diamond Lemma, which had been published by Newman in 1942 and turned to be very useful in algebra and functional analysis. We replace his confluence condition by so-called mediator condition, which has a clear topological meaning. Using this new Diamond Lemma, we get several interesting results:
1. The Kneser-Milnor prime decomposition theorem (new proof).
2. The Swarup theorem for boundary connected sums for orientable 3-manifolds (new proof and generalization two non-orientable case).
3. A spherical splitting theorem for knotted graphs in 3-manifolds.
4. Counterexamples to the prime decomposition theorem for 3-orbifolds. During a long time the uniqueness of prime decompositions of 3-orbifolds had been accepted by mathematical community as a folklore theorem. So the existence of counterexamples is quite unexpected.
5. A new theorem on annular splitting of 3-manifolds, which is independent of the JSJ-decomposition theorem.
6. Prime decomposition theorem for virtual knots (new result).

14. MaiVladimir Chernov (Dartmouth)Low conjecture, Linking and causality in globally hyperbolic Lorentz spacetimesLow conjecture and the Legendrian Low conjecture formulated by Natário and Tod say that for many spacetimes X two events x,y in X are causally related if and only if the link of spheres S_x, S_y whose points are light rays passing through x and y is non-trivial in the contact manifold N of all light rays in X.
We prove the Low and the Legendrian Low conjectures and show that similar statements are in fact true in almost all 4-dimensional globally hyperbolic spacetimes.
We also show that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R^4.

16. August(Sondertermin14:00-17:00, SR 3517 EAP)Michael Hinz(University ofConneticut) (14:00-15:00)"Vector analysis on fractals and applications"Abstract: The talk is concerned with some recent progress in the analysis on fractals. We discuss the construction of a vector analysis that is solely based on the notion of (Dirichlet) energy. It is related to differential geometry (L²-differential forms) as well as to stochastic analysis (additive functionals of Markov processes) and can be used to study scalar and vector equations on fractals, for example magnetic Schrödinger equations or Navier-Stokes type models.Radoslaw Wojciechowski (York College New York)(15:00-16:00)"Uniqueness of self-adjoint extensions of graph Laplacians"Xueping Huang (Bielefeld University)(16:00-17:00)"An analytic approach to stochastic completeness of weighted graphs"

11. JuliKlaus Mecke(Erlangen)"Tensor valuations: linking physics to spatial structures"Abstract: Spatially structured matter such as foams, gels or biomaterials are of increasing technological importance due to their shape-dependent material properties. But the shape of disordered structures is a remarkably incoherent concept and cannot be captured by correlation functions alone. Integral geometry furnishes a suitable family of morphological descriptors, so-called tensorial Minkowski functionals, which are related to curvature integrals and do not only characterize shape but also anisotropy and even topology of disordered structures. These measures can be used to derive structure-property relations for complex materials.

4. JuliJun Morita (Tsukuba)"Symmetry, Lie algebras and aperiodic orders"Abstract: A common key word between Lie theory and aperiodic theory is symmetry. We review several recent topics in both areas including Moody conjecture and Kac conjecture as well as a new invariant of words and automata. Then, we create a new approach combining Lie structures and aperiodic structures.

2.MaiJan Rataj (Prague)"Normal cycles and curvature measures for sets with d.c. boundary"Abstract: A real function (defined on an open convex subset of $R^d$) is called d.c. if it can be written as a difference of two convex functions. Let $A$ be a $d$-dimensional d.c. submanifold in $R^d$, i.e., each point of $A$ has a neighbourhood which can be represented as a subgraph of a d.c. function. We show that $A$ admits a normal cycle and, hence, its curvature measures can be defined so that the Gauss-Bonnet formula holds.

25. AprilBatu Güneysu (HU Berlin)"Hydrogen type stability problems on manifolds"Abstract: In this talk, I will explain how classical results on the stability of Hydrogen type atoms can be extended to certain abstract Riemannian 3-manifolds. This clarifies which geometric and topological properties of the Euclidean space are actually needed to formulate and prove such stability results. If time admits, I will also explain some path integral techniques for the underlying Schrödinger semigroups.

19. April (10:15-11:45, SR 207 CZ)Naotaka Kajino (Bielefeld)"Weyl's Laplacian eigenvalue asymptotics for the measurable Riemannian structure on the Sierpinski gasket"Abstract: On the Sierpinski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the "gradient vector fields" of functions, the "Riemannian volume measure" and the "geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bound for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal. 36 (2012), 67--115]. In the talk, Weyl's Laplacian eigenvalue asymptotics is presented for this case. The correct scaling order for the asymptotics of the eigenvalues is given by the Hausdorff dimension d of the gasket with respect to the "geodesic metric", and in the limit of the eigenvalue asymptotics we obtain a constant multiple of the d-dimensional Hausdorff measure. Moreover, we will also see that this Hausdorff measure is Ahlfors regular with respect to the "geodesic metric" but that it is singular to the "Riemannian volume measure".

Naotaka Kajino (Bielefeld)(17:15-18:15)"On-diagonal oscillation of the heat kernels on self-similar fractals"Abstract: For the canonical heat kernel $p_{t}(x,y)$ associated with a self-similar Dirichlet form on a self-similar fractal, the following recent results of the speaker will be presented: under certain mild assumptions on the fractal or on $p_{t}(x,y)$, for a "generic" (in particular, almost every) point $x$ of the fractal, $p_{(\cdot)}(x,x)$ NEITHER varies regularly at $0$ (and hence $t^{a}p_{t}(x,x)$ does NOT converge for the appropriate scaling order $a$) NOR admits a log-periodic oscillation as $t\downarrow 0$. Furthermore the non-existence of the limit of $t^{a}p_{t}(x,x)$ is proved for ANY point $x$ of the fractal in the particular cases of the $d$-dimensional standard Sierpinski gasket with $d\geq 2$ and of the $N$-polygasket with $N\geq 3$ odd, e.g. the pentagasket ($N=5$) and the heptagasket ($N=7$).

Seminarplan WS 2011/12

15. Februar 16:15 (Seminar Analysis- Ernst-Abbe-Platz 2, SR 3519)Mark Lawson (Edingburgh)"From partial symmetries to non-commutative Stone duality"Abstract： Shechtman's work on quasi-crystals, for which he received the 2011 Nobel prize for chemistry, inspired both mathematicians and physicists to investigate more deeply the specific theory of aperiodic tilings, but it also raised more general questions about the nature of symmetry and how it can be formalized mathematically. In this talk, I shall describe one way, inverse semigroups, in which the classical notion of group has been extended to deal with more exotic notions of symmetry. I shall explain how this theory had its origins in the the classical work of Lie, was then developed in the mid 1950's only to be rather marginalized, and then experienced a renaissance in the 1990's with the discovery of unexpected connections with aperiodic tilings and C*-algebras. My main goal in this talk is to explain one way of generalizing Marshall Stone's famous duality between Boolean algebra and Boolean spaces that uses inverse semigroups --- and why this might be interesting. The work I shall describe evolved, and continues to develop, in collaboration with Jonathon Funk (West Indies), Johannes Kellendonk (Lyon), Ganna Kudryavtseva (Ljublyana), Daniel Lenz (Jena), Stuart Margolis (Bar Ilan), and Ben Steinberg (CUNY).

01. Februar 16:15Jiaxin Hu (Tsinghua University Bejing)"Heat kernel estimates for non-local Dirichlet forms on metric spaces"Abstract：I will present new heat kernel upper bounds for a certain class of non-local regular Dirichlet forms on metric measure spaces, including fractal spaces. As an application, we obtain two-sided estimates of heat kernels for non-local regular Dirichlet forms with finite effective resistance, including settings with the walk dimension greater than 2. The talk is based on a joint work with Alexander Grigor'yan and Ka-Sing Lau.

7.DezemberPeter Stollmann (TU Chemnitz)"The complex Laplacian and its heat semigroup"The $\bar{\partial}$ Neumann operator is an important operator from several complex variables. In joint work with J. Perez we study its heat semigroup in noncompact situations. In the talk we give an overview over these results.

30. November,Ivan Veselic, (Technische Universität Chemnitz)TBA

23. NovemberHolger R. Dullin (University of Sydney)"A Lie-Poisson structure and integrator for the reduced N-body problem"The general N-body problem is invariant under the symmetry group of translations, rotations, and Galilein boosts. The Hilber-Weyl invariants of this symmetry group can be represented by symmetric block-Laplacian matrices and we show that they satisfy a Lie-Poisson structure. Using this Lie-Poisson structure we construct a splitting integrator for the symmetry reduced N-body problem. For small N=3,4 this gives an efficient computational method, which is illustrated by computing the figure-8 choreography orbit in 3 steps.

16.November
Shiping Liu (MPI Leipzig)"Ollivier-Ricci curvature on neighborhood graphs"Abstract: In this talk, I will firstly give two kinds of understanding of Ollivier-Ricci curvature on graphs. One is the relation with local clustering, or number of triangles and self-loops. The other one is the relation with the expectation distance between two random walks. Based on those two aspects, I will discuss the combination of this curvature and the so-called neighborhood graph method which is developed by Bauer-Jost in order to explore the eigenvalue estimates of normalized graph Laplace operator.